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Father–3 equivalence continuum: Difference between revisions

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Rework into a more typical definition of n. The old n is now k.
Remove the k-continuum since no one is actively arguing for it. Also remove the 3 & 33c temp, which is unenlighted result of looking at the continuum that way
 
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The '''chromatic-diatonic equivalence continuum''', despite its name, is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[16/15|classical diatonic semitones (16/15)]] with the [[32/27|Pythagorean minor third (32/27)]].
The '''father–3 equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[16/15|classical diatonic semitones (16/15)]] with the [[32/27|Pythagorean minor third (32/27)]].


All temperaments in the continuum satisfy (16/15)<sup>''n''</sup> ~ 32/27. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.  
Note that because 3et is a record equal temperament in the [[2.5 subgroup]], the continuum can be conceptualized as the [[Father–3 equivalence continuum/Godtone's approach|''augmented–dicot equivalence continuum'']], which Godtone argues is easier to understand, with characteristic 2.5-subgroup [[comma]] [[128/125]] as the interval with a single factor of 3 is [[25/24]].


32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, we notice that ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|3]] in the generator chain. Such an equivalence continuum is more properly called the ''father-3 equivalence continuum''.  
All temperaments in the continuum satisfy {{nowrap|(16/15)<sup>''n''</sup> ~ 32/27}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[father]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[3edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is approximately 2.63252…, and temperaments having ''n'' near this value tend to be the most accurate ones.
 
32/27 is the characteristic 3-limit comma tempered out in 3edo. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|3]] in the generator chain.  


{| class="wikitable center-1"
{| class="wikitable center-1"
|+ Temperaments with integer ''n''
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''n''
! rowspan="2" | ''n''
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|-
|-
| 0
| 0
| [[Alteraugment]]
| [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
| [[32/27]]
| [[32/27]]
| {{monzo| 5 -3 }}
| {{Monzo| 5 -3 }}
|-
|-
| 1
| 1
| [[Very low accuracy temperaments #Yellow (2c&3)|Yellow]]
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
| {{Monzo| 1 -2 1 }}
|-
|-
| 2
| 2
| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
| {{Monzo| -3 -1 2 }}
|-
|-
| 3
| 3
| [[Augmented]]
| [[Augmented (temperament)|Augmented]]
| [[128/125]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
| {{Monzo| 7 0 -3 }}
|-
|-
| 4
| 4
| [[Smate]]
| [[Smate]]
| [[2048/1875]]
| [[2048/1875]]
| {{monzo| 11 -1 -4 }}
| {{Monzo| 11 -1 -4 }}
|-
|-
| …
| …
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| [[Father]]
| [[Father]]
| [[16/15]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| {{Monzo| 4 -1 -1 }}
|}
|}


We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''yellow-3 equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.61255…  
We may invert the continuum by setting ''m'' such that {{nowrap| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''antonian–3 equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.61255…  


{| class="wikitable center-1"
{| class="wikitable center-1"
|+ Temperaments with integer ''m''
|+ style="font-size: 105%;" | Temperaments with integer ''m''
|-
|-
! rowspan="2" | ''m''
! rowspan="2" | ''m''
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|-
|-
| 0
| 0
| [[Alteraugment]]
| [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
| [[32/27]]
| [[32/27]]
| {{monzo| 5 -3 }}
| {{Monzo| 5 -3 }}
|-
|-
| 1
| 1
| [[Father]]
| [[Father]]
| [[16/15]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| {{Monzo| 4 -1 -1 }}
|-
|-
| 2
| 2
| [[Dicot]]
| [[Dicot]]
| [[25/24]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
| {{Monzo| -3 -1 2 }}
|-
|-
| …
| …
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|-
|-
| ∞
| ∞
| [[Very low accuracy temperaments #Yellow (2c&3)|Yellow]]
| [[Very low accuracy temperaments #Antonian|Antonian]]
| [[10/9]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
| {{Monzo| 1 -2 1 }}
|}
|}


{| class="wikitable"
{| class="wikitable"
|+ Temperaments with fractional ''n'' and ''m''
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
|-
! Temperament !! ''n'' !! ''m''
|-
| [[Wesley]] || 7/3 = 2.{{overline|3}} || 7/4 = 1.75
|-
| [[Magic]] || 5/2 = 2.5 || 5/3 = 1.{{overline|6}}
|-
|-
| [[Würschmidt]] || 8/3 = 2.{{overline|6}} || 8/5 = 1.6
! ''n'' !! ''m''!! Temperament || Comma
|-
|-
| [[Isnes]] || 19/7 = 2.{{overline|714285}} || 19/12 = 1.58{{overline|3}}
| 7/3 = 2.{{overline|3}} || 7/4 = 1.75 || [[Wesley]] || {{monzo| -13 -2 7 }}
|-
| [[Magus]] || 11/4 = 2.75 || 11/7 = 1.{{overline|571428}}
|}
 
Some prefer conceptualizing this continuum in terms of ''k'' = 1/(''n'' - 2) such that temperaments satisfy (25/24)<sup>''k''</sup> = 16/15. This is the source of the name ''chromatic-diatonic equivalence continuum'', where both ''chromatic'' and ''diatonic'' refer to the classical versions of semitones. The just value of ''k'' is approximately 1.58097…
 
{| class="wikitable center-1"
|+ Temperaments with integer ''k''
|-
! rowspan="2" | ''k''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| -1
| [[Very low accuracy temperaments#Yellow (2c&3)|Yellow]]
| [[10/9]]
| {{monzo| 1 -2 1 }}
|-
| 0
| [[Father]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
|-
|-
| 1
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Magic]] || {{monzo| -10 -1 5 }}
| [[Augmented]]
| [[128/125]]
| {{monzo| 7 0 -3 }}
|-
|-
| 2
| 21/8 = 2.625 || 21/13 = 1.{{overline|615384}} || [[Mutt]] || {{monzo| -44 -3 21 }}
| [[Magic]]
| [[3125/3072]]
| {{monzo| 10 1 -5 }}
|-
|-
| 3
| 29/11 = 2.{{overline|63}} || 29/18 = 1.6{{overline|1}} || [[Squarschmidt]] || {{monzo| 61 4 -29 }}
| [[Wesley family|Wesley]]
| 78125/73728
| {{monzo| 13 2 -7 }}
|-
|-
| 4
| 8/3 = 2.{{overline|6}} || 8/5 = 1.6 || [[Würschmidt]] || {{monzo| 17 1 -8 }}
| 3 & 33c
| 1953125/1769472
| {{monzo| 16 3 -9 }}
|-
|-
| …
| 19/7 = 2.{{overline|714285}} || 19/12 = 1.58{{overline|3}} || [[#Isnes|Isnes]] || {{monzo| 41 2 -19 }}
| …
| …
| …
|-
|-
| ∞
| 11/4 = 2.75 || 11/7 = 1.{{overline|571428}} || [[Magus]] || {{monzo| 24 1 -11 }}
| [[Dicot]]
| [[25/24]]
| {{monzo| -3 -1 2 }}
|}
|}


== 3 & 33c ==
== Mutt (5-limit) ==
{{Main| Mutt }}
: ''For extensions, see [[Horwell temperaments #Mutt]].''


Comma list: {{monzo| 16 3 -9 }}
[[Subgroup]]: 2.3.5


POTE generator: 34.0971 cents
[[Comma list]]: {{monzo| -44 -3 21 }}


Mapping: [{{val| 3 5 7 }}, {{val| 0 -3 -1 }}]
{{Mapping|legend=1| 3 -2 6 | 0 7 1 }}
: mapping generators: ~98304/78125, ~5/4


{{Optimal ET sequence|legend=1| 3, 6, 9b, 33c }}
[[Optimal tuning]]s:
* [[WE]]: ~98304/78125 = 400.0227{{c}}, ~5/4 = 386.0017{{c}} (~393216/390625 = 14.0210{{c}})
: [[error map]]: {{val| +0.068 +0.012 -0.176 }}
* [[CWE]]: ~98304/78125 = 400.0000{{c}}, ~5/4 = 385.9858{{c}} (~393216/390625 = 14.0142{{c}})
: error map: {{val| 0.000 -0.055 -0.328 }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=3_33c&limit=5 The temperament finder - 5-limit 3 & 33c]
{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455, 4194cc, 5649cc }}
 
[[Badness]] (Sintel): 3.81


== Isnes ==
== Isnes ==
Isnes is so called because the generator is half of a [[5/2]] major tenth, in a similar way that [[sensi]] has a generator of half a [[5/3]] major sixth. This corresponds to {{nowrap|''n'' {{=}} 19/7 }} and {{nowrap| ''m'' {{=}} 19/12 }}.
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 41 2 -19 }}
{{Mapping|legend=1| 1 -11 1 | 0 19 2 }}
: mapping generators: ~2, ~3145728/1953125
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2782{{c}}, ~3145728/1953125 = 794.4174{{c}}
: [[error map]]: {{val| -0.722 -0.090 +1.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3145728/1953125 = 794.8728{{c}}
: error map: {{val| 0.000 +0.628 +3.432 }}
{{Optimal ET sequence|legend=1| 3, 71b, 74, 77, 157, 548ccc }}
[[Badness]] (Sintel): 30.4
== Squarschmidt (5-limit) ==
: ''For extensions, see [[Hemimage temperaments #Squarschmidt]].''
A generator for the squarschmidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents.


So called because the generator is half of a [[8/5]] minor sixth, in a similar way that [[sensi]] has a generator of half a [[5/3]].
[[Subgroup]]: 2.3.5


Comma list: {{Monzo|41 2 -19}}
[[Comma list]]: {{monzo| 61 4 -29 }}


POTE generator: 12582912/9765625 ~ 1953125/1572864 = 405.1047 cents
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
: mapping generators: ~2, ~98304/78125


Mapping: [{{val| 1 8 3 }}, {{val| 0 -19 -2 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9653{{c}}, ~98304/78125 = 396.6094{{c}}
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98304/78125 = 396.6201{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}


{{Optimal ET sequence|legend=1| 3, 74, 77, 80, 83, 154, 157, 160 }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947, 9588cc, 10535cc, 11482ccc }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=3_77&limit=5 The temperament finder - 5-limit 3 & 77]
[[Badness]] (Sintel): 5.12


[[Category:3edo]]
[[Category:3edo]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]
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